pina-3d

name	pina-3d
description	Guidance for composing 3D PINA problems — sampling budgets, activation choices, and gotchas specific to three spatial axes plus optional time.
triggers	["3d problem","three dimensional pde","volumetric pde","3d poisson","3d heat","3d navier-stokes","lid-driven cavity","3d wave"]

PINA 3D Problems

3D is the engineering default for this project. The composer already handles arbitrary axis sets — {"x": [...], "y": [...], "z": [...]} produces a 3D SpatialProblem with zero Python changes; adding "t": [...] promotes to TimeDependentProblem. This skill captures the scaling + numerical considerations that only become important once you leave 2D.

Composition — no surprises

spec = ProblemSpec(
    name="heat_3d",
    output_variables=["u"],
    domain_bounds={
        "x": [0.0, 1.0], "y": [0.0, 1.0], "z": [0.0, 1.0],
        "t": [0.0, 1.0],
    },
    subdomains=[
        SubdomainSpec(name="D", bounds={
            "x": [0.0, 1.0], "y": [0.0, 1.0], "z": [0.0, 1.0], "t": [0.0, 1.0],
        }),
        # Faces: pin exactly one axis
        SubdomainSpec(name="x0", bounds={
            "x": 0.0, "y": [0,1], "z": [0,1], "t": [0,1]
        }),
        # … and so on for the other five walls.
    ],
    equations=[
        EquationSpec(
            name="heat",
            form="u_t - alpha*(u_xx + u_yy + u_zz)",
            outputs=["u"],
            derivatives=[
                DerivativeSpec(name="u_t",  field="u", wrt=["t"]),
                DerivativeSpec(name="u_xx", field="u", wrt=["x","x"]),
                DerivativeSpec(name="u_yy", field="u", wrt=["y","y"]),
                DerivativeSpec(name="u_zz", field="u", wrt=["z","z"]),
            ],
            parameters={"alpha": 0.1},
        ),
    ],
    conditions=[
        ConditionSpec(subdomain="D", kind="equation", equation_name="heat"),
        # Dirichlet walls via fixed_value; IC via a t=0 subdomain + equation_inline.
    ],
)

Sampling budgets

A 2D Poisson trains well on ~1k collocation points; 3D needs 10x that just to cover the volume evenly. Rule of thumb:

• 3D steady-state: n_points = 8000–16000.
• 3D + time: n_points = 20000–40000.
• Use sample_mode="latin" — uniform random gets clustery quickly in 3D.

Architecture tuning

• Minimum hidden width: 64. The 8×8 FeedForward that trains 1D Burgers in 2 epochs will not fit 3D without adjustment.
• Depth: 5–8 layers; deeper helps higher-frequency features but Adam stalls earlier, use LBFGS for the last 20% of training.
• Activations: tanh or silu. Avoid relu — its second derivative is zero, which nulls the viscous / diffusive terms.
• For Navier-Stokes 3D, Fourier Feature Nets (a.k.a. RFF) beat plain FeedForward. ModelManager.create("fno", ...) is the right first pick.

Visualisation

The provenance dashboard (Phase C3) uses pyvista for 3D rendering. Export inference output via Trainer.test() on a volumetric grid (e.g. 32³) and load the resulting numpy array into a pyvista.ImageData. Until that's wired, fall back to 2D slices — the 2D viz in examples/02_provenance_dashboard.py takes (x, y, value) at fixed z, t.

Gotchas

• PINA's input_variables on a 3D+time problem is ["x","y","z","t"] — verify with cls().input_variables after composing.
• Derivative operators scale with the number of axes, so the residual cost grows as O(d²) for a full Laplacian. Do not compose a 3D problem with Δu + … as a single wrt=["x","x"]; list each direction as its own derivative so the operator graph stays shallow.
• Memory: one 40k × (3+1+hidden_width) float32 tensor is ~13 MB per batch. Watch GPU memory if you push past 100k points.